QuestionThe function $f(x)=\frac{1}{x+4}-3$ is a rational function. a. Use transformations of $y=\frac{1}{x}$ or $y=\frac{1}{x^{2}}$ to sketch the graph. b. Find all $x$-intercepts or state that the function has no $x$-intercepts. c. Find the $y$-intercept or state that the function does not have a $y$-intercept. d. Find the equation(s) of all vertical asymptotes. e. Find the equation(s) of all horizontal asymptotes.

Studdy Solution
Identify horizontal asymptotes by considering the behavior as $x \to \pm \infty$:
As $x \to \pm \infty$, $\frac{1}{x+4} \to 0$.
Thus, $f(x) = \frac{1}{x+4} - 3 \to -3$.
Therefore, the horizontal asymptote is $y = -3$.
The graph transformations, intercepts, and asymptotes have been determined as follows: - Transformations: Left 4 units, down 3 units. - $x$-intercept: $\left(-\frac{11}{3}, 0\right)$. - $y$-intercept: $\left(0, -\frac{11}{4}\right)$. - Vertical asymptote: $x = -4$. - Horizontal asymptote: $y = -3$.